WEBVTT
00:00:00.360 --> 00:00:07.640
In this video, we will learn how to use the Pythagorean identities to find the values of trigonometric functions.
00:00:08.280 --> 00:00:11.720
We will begin by recalling some key formulae and definitions.
00:00:12.440 --> 00:00:18.040
We know that the trig functions are periodic and have an infinite number of solutions.
00:00:19.120 --> 00:00:25.280
However, in this video, we will focus on solutions between zero and 360 degrees using the CAST diagram.
00:00:26.200 --> 00:00:30.240
The π₯- and π¦-axes create four quadrants as shown.
00:00:31.280 --> 00:00:51.920
Quadrant one has positive π₯- and positive π¦-values; quadrant two has negative π₯- and positive π¦-values; quadrant three, negative π₯ and negative π¦; and quadrant four, positive π₯ and negative π¦.
00:00:52.880 --> 00:01:01.560
When dealing with our trig functions, we measure in a counterclockwise direction from the positive π₯-axis.
00:01:02.200 --> 00:01:13.960
This means that quadrant one contains angles between zero and 90 degrees; quadrant two, between 90 and 180 degrees; and so on.
00:01:14.720 --> 00:01:18.520
We can label the quadrants with the acronym CAST as shown.
00:01:19.040 --> 00:01:24.920
In quadrant one, labeled A, the sin of angle π, cos of angle π, and tan of angle π are all positive.
00:01:25.480 --> 00:01:28.640
In quadrant two, the sin of angle π is positive.
00:01:29.280 --> 00:01:32.720
However, the cos of angle π and tan of angle π are both negative.
00:01:33.560 --> 00:01:47.520
In quadrant three, when π lies between 180 and 270 degrees, the tan of π is positive, whereas the sin of π and cos of π are both negative.
00:01:48.120 --> 00:01:55.320
Finally, in quadrant four, the cos of π is positive and the sin and tan of π are negative.
00:01:56.320 --> 00:02:02.280
In this video, we will also need to recall the three reciprocal trig identities.
00:02:03.120 --> 00:02:06.240
The csc of angle π is equal to one over the sin of angle π.
00:02:06.720 --> 00:02:09.760
The sec of angle π is equal to one over the cos of angle π.
00:02:10.240 --> 00:02:13.640
And the cot of angle π is equal to one over the tan of angle π.
00:02:14.440 --> 00:02:24.000
If the functions sin π, cos π, and tan π are positive or negative, then their reciprocal will also be positive or negative.
00:02:24.640 --> 00:02:28.400
This means that all six functions are positive in the first quadrant.
00:02:29.040 --> 00:02:40.520
In the second quadrant, between 90 and 180 degrees, the sin of π and csc of π will be positive, whereas all four other functions will be negative.
00:02:41.080 --> 00:02:44.400
This pattern continues in quadrant three and four.
00:02:45.000 --> 00:02:54.960
In the questions that follow in this video, we will need to use this information to decide whether our answers are positive or negative.
00:02:55.520 --> 00:02:59.080
We will now consider the three Pythagorean identities.
00:02:59.560 --> 00:03:04.720
Firstly, we have sin squared π plus cos squared π is equal to one.
00:03:05.360 --> 00:03:12.000
Our second identity is tan squared π plus one is equal to sec squared π.
00:03:12.680 --> 00:03:25.400
Whilst we donβt need to consider the proofs of the identities in this video, we can get from the first identity to the second identity by dividing each term by cos squared π.
00:03:26.240 --> 00:03:35.480
sin squared π divided by cos squared π is tan squared π, as sin π over cos π equals tan π.
00:03:35.960 --> 00:03:40.800
Dividing cos squared π by cos squared π gives us one.
00:03:41.680 --> 00:03:48.680
From the reciprocal identities, we know that one divided by cos squared π is sec squared π.
00:03:49.360 --> 00:03:56.520
The third Pythagorean identity is one plus cot squared π is equal to csc squared π.
00:03:57.280 --> 00:04:03.680
This can be obtained by dividing each term of the first identity by sin squared π.
00:04:04.280 --> 00:04:08.480
sin squared π divided by sin squared π is equal to one.
00:04:09.280 --> 00:04:14.840
cos squared π divided by sin squared π is equal to one over tan squared π.
00:04:15.480 --> 00:04:17.720
And this is equal to cot squared π.
00:04:18.320 --> 00:04:23.520
Finally, one divided by sin squared π is equal to csc squared π.
00:04:24.520 --> 00:04:29.960
We will now use these three identities together with the CAST diagram to solve some trig equations.
00:04:30.560 --> 00:04:44.080
Find cos π given sin π is equal to negative three-fifths, where π is greater than or equal to 270 degrees and less than 360 degrees.
00:04:44.880 --> 00:04:47.640
There are lots of ways of solving this problem.
00:04:48.280 --> 00:04:56.160
In this video, we will use the Pythagorean identity sin squared π plus cos squared π is equal to one.
00:04:56.760 --> 00:05:09.280
Before substituting in our value of sin π, it is worth noting that π must be greater than or equal to 270 degrees and less than 360 degrees.
00:05:09.680 --> 00:05:17.480
Using our knowledge of the CAST diagram, this means that π must lie in the fourth quadrant.
00:05:18.040 --> 00:05:24.680
In this quadrant, the cos of angle π is positive, whereas the sin of π and tan of π are negative.
00:05:25.240 --> 00:05:30.640
This ties in with the fact that we are told that sin π is negative three-fifths.
00:05:31.240 --> 00:05:36.440
We know that our answer for cos of π must be positive.
00:05:37.080 --> 00:05:41.640
We can now substitute the value of sin π into our Pythagorean identity.
00:05:42.320 --> 00:05:48.960
This gives us negative three-fifths squared plus cos squared π is equal to one.
00:05:50.040 --> 00:05:53.920
Squaring a negative number gives a positive answer.
00:05:54.400 --> 00:05:59.800
Therefore, negative three-fifths squared is equal to nine twenty-fifths.
00:06:00.800 --> 00:06:04.440
We can then subtract this from both sides of our equation.
00:06:05.080 --> 00:06:08.840
cos squared π is equal to 16 over 25 or sixteen twenty-fifths.
00:06:09.600 --> 00:06:19.960
Square rooting both sides of this equation, we see that cos of π is equal to positive or negative the square root of 16 over 25.
00:06:20.760 --> 00:06:28.160
When square rooting a fraction, we simply square root the numerator and denominator separately.
00:06:28.640 --> 00:06:34.920
The square root of 16 is four, and the square root of 25 is five.
00:06:35.320 --> 00:06:42.480
As the cos of π must be positive, we can conclude that the cos of π is four-fifths.
00:06:42.920 --> 00:06:50.880
In our next question, we need to evaluate the sine function given the cosine function and the quadrant of an angle.
00:06:51.840 --> 00:07:03.840
Find the value of sin π given cos of π is equal to negative 21 over 29, where π is greater than 90 degrees and less than 180 degrees.
00:07:04.720 --> 00:07:13.360
In order to answer this question, weβll use the Pythagorean identity sin squared π plus cos squared π is equal to one.
00:07:14.160 --> 00:07:23.600
We are told that π lies between 90 and 180 degrees, so it is worth considering our CAST diagram.
00:07:24.320 --> 00:07:30.360
The angle lies in the second quadrant, which means that the sin of angle π must be positive.
00:07:31.120 --> 00:07:35.800
The cos of angle π and tan of angle π must be negative.
00:07:36.320 --> 00:07:42.920
This ties in with the fact that we are told that the cos of π is equal to negative 21 over 29.
00:07:43.720 --> 00:07:48.920
It also helps us in that we know the answer for sin π must be positive.
00:07:49.400 --> 00:07:53.200
We can substitute the value of cos π into the Pythagorean identity.
00:07:53.760 --> 00:08:01.680
Squaring negative 21 over 29 gives us 441 over 841.
00:08:02.400 --> 00:08:13.080
We can then subtract this from both sides of our equation such that sin squared π is equal to one minus 441 over 841.
00:08:13.720 --> 00:08:18.440
The right-hand side simplifies to 400 over 841.
00:08:19.200 --> 00:08:31.600
We can then square root both sides of our equation so that sin of π is equal to positive or negative the square root of 400 over 841.
00:08:32.440 --> 00:08:39.840
Square rooting the numerator gives us 20 and the denominator 29.
00:08:40.560 --> 00:08:55.200
As sin of π must be positive, if the cos of π is negative 21 over 29 and π lies between 90 and 180 degrees, then the sin of π is equal to 20 over 29.
00:08:55.960 --> 00:09:01.800
In our next question, we will use the Pythagorean identities to evaluate an expression.
00:09:02.560 --> 00:09:09.240
Find the value of sin π cos π given sin π plus cos π is equal to five-quarters.
00:09:09.960 --> 00:09:14.200
In order to solve this problem, we will need to recall the Pythagorean identities.
00:09:14.720 --> 00:09:19.760
We will begin with the equation sin π plus cos π is equal to five over four.
00:09:20.800 --> 00:09:23.280
We can square both sides of this equation.
00:09:23.760 --> 00:09:29.720
The left-hand side becomes sin π plus cos π multiplied by sin π plus cos π.
00:09:30.160 --> 00:09:37.360
The right-hand side is equal to 25 over 16 as we simply square the numerator and denominator separately.
00:09:38.040 --> 00:09:52.760
Distributing the parentheses or expanding the brackets using the FOIL methods, we get sin squared π plus sin π cos π plus sin π cos π plus cos squared π.
00:09:53.640 --> 00:09:55.880
We can group or collect the middle terms.
00:09:56.440 --> 00:10:07.280
sin squared π plus two sin π cos π plus cos squared π is equal to 25 over 16.
00:10:08.080 --> 00:10:15.000
One of the Pythagorean identities states that sin squared π plus cos squared π is equal to one.
00:10:15.920 --> 00:10:23.960
This means we can rewrite the left-hand side of our equation as two sin π cos π plus one.
00:10:24.840 --> 00:10:28.480
We can then subtract one from both sides of this equation.
00:10:29.120 --> 00:10:34.600
25 over 16 minus one is equal to nine over 16.
00:10:35.400 --> 00:10:44.800
We can then divide both sides of this new equation by two, giving us sin π cos π is equal to nine over 32.
00:10:45.480 --> 00:10:54.440
If sin π plus cos π is equal to five over four, then sin π multiplied by cos π is equal to nine over 32.
00:10:55.360 --> 00:11:01.400
In our final question, we will need to use a different Pythagorean identity.
00:11:02.320 --> 00:11:11.560
Find the sec of π minus the tan of π given the sec of π plus the tan of π is equal to negative 14 over 27.
00:11:12.760 --> 00:11:22.000
We recall that the difference of two squares states that π₯ squared minus π¦ squared is equal to π₯ plus π¦ multiplied by π₯ minus π¦.
00:11:22.960 --> 00:11:33.320
This means that sec π plus tan π multiplied by sec π minus tan π is equal to sec squared π minus tan squared π.
00:11:33.760 --> 00:11:37.560
We are told in the question the value of sec π plus tan π.
00:11:38.320 --> 00:11:41.600
It is equal to negative 14 over 27.
00:11:42.440 --> 00:11:46.000
And we need to calculate the value of sec π minus tan π.
00:11:46.960 --> 00:11:53.640
One of the Pythagorean identities states that tan squared π plus one is equal to sec squared π.
00:11:54.400 --> 00:12:03.720
If we subtract tan squared π from both sides of this equation, we have sec squared π minus tan squared π is equal to one.
00:12:04.360 --> 00:12:06.920
This is equivalent to the right-hand side of our equation.
00:12:07.560 --> 00:12:13.920
Negative 14 over 27 multiplied by sec π minus tan π is equal to one.
00:12:14.520 --> 00:12:20.280
We can then divide both sides of this equation by negative 14 over 27.
00:12:21.000 --> 00:12:26.000
We know that dividing by a fraction is the same as multiplying by the reciprocal of this fraction.
00:12:26.600 --> 00:12:32.840
Therefore, the right-hand side is equal to one multiplied by negative 27 over 14.
00:12:33.320 --> 00:12:44.920
If the sec of π plus the tan of π is equal to negative 14 over 27, then the sec of π minus the tan of π is equal to negative 27 over 14.
00:12:45.800 --> 00:12:49.080
These values are the reciprocal of one another.
00:12:50.040 --> 00:12:54.240
We will now summarize the key points from this video.
00:12:54.880 --> 00:12:57.960
The three Pythagorean identities are as follows.
00:12:58.920 --> 00:13:02.520
sin squared π plus cos squared π is equal to one.
00:13:03.280 --> 00:13:06.880
tan squared π plus one is equal to sec squared π.
00:13:07.680 --> 00:13:11.400
One plus cot squared π is equal to csc squared π.
00:13:12.160 --> 00:13:19.160
We have seen in this video that we can use these identities to find the values of trig functions.
00:13:19.720 --> 00:13:30.480
When solving any trig equations, it is also important to recall which of the functions are positive and negative in each quadrant.
00:13:30.880 --> 00:13:34.120
One way of doing this is using the CAST diagram as shown.